Long post alert...
Either car is a good choice. I chose to get a 300ZX TT because I didn't want to have to swap the motor just to have a fast car. The 300ZX is hard to work on, but it's actually easy to modify if you aren't looking for giant HP. People have run 11's on the stock turbos. My car, just throwing some bolt-ons at it, (intercooler, intake, full exhaust, chip, boost controller, bigger injectors) and I have run 12.2 at 116 mph with no dyno tuning and very little drag racing experience.
As for handling, the 240sx is lighter, so it has an inherit advantage. The 300ZX understeers in tight corners, but you can dial that out with suspension mods and tuning. Like a Supra, you simple have to counter the weight with big wide tires, which thankfully the car has appropriate fenders for.
The biggest problem with the 300ZX is really the weight... the biggest attraction for me was the durability of the car. The engine is very strong, and can make Supra rivaling power 800+ rwhp, although its harder to extract that power compared to a Supra. Unlike a SR20DET, the block itself is extremely strong. The rear end is very strong also, hard drag launches are no problem. Some guys have even run 1.5 sec 60 fts.
However, to me the 300zx has more heritage. It won its class at 24hrs of Lemans in '94, won 24hrs of Daytona, Sebring 12 hour victories, not to mention that Jun set a record of 260 mph at Bonneville. The 240SX has, um, plenty of drifting victories. But the 300ZX has at least 1 D1 victory, and I have the Japanese magazine article to prove it!
So both are good cars, the main attraction of a 240SX to me is it's lightweight and relative simplicity, the 300ZX has power and durability, at the cost of weight and complexity.